SOLUTION: The acute angle between the line x-2y+3=0 and the line y=mx is 45 degrees. Show that absolute(2m-1/m+2)=1 and find all possible values of m

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Question 1140357: The acute angle between the line x-2y+3=0 and the line y=mx is 45 degrees. Show that absolute(2m-1/m+2)=1 and find all possible values of m
Answer by ikleyn(52903) About Me  (Show Source):
You can put this solution on YOUR website!
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The line x - 2y + 3 = 0  has the slope-intercept form  y = x%2F2+%2B+3%2F2  and has, therefore, the slope  1%2F2.


So, it makes the angle  alpha  with the horizontal axis "x"  such that  tan%28alpha%29 = 1%2F2.                   (1)


The line  y = mx  makes the angle  beta  with the horizontal axis "x"  such that  tan%28beta%29 = m.         (2)


The angle  alpha - beta is equal to 45 degree, according to the condition.


Hence,  abs%28tan%28alpha-beta%29%29 = 1.


By applying the formula for tangent of the difference of arguments, you will get then


    1 = abs%28tan%28alpha-beta%29%29 = abs%28%28tan%28alpha%29+-+tan%28beta%29%29%2F%281+%2B+tan%28alpha%29%2Atan%28beta%29%29%29 = substitute here (1) and (2) to get = abs%28%281%2F2+-+m%29%2F%281%2B%281%2F2%29%2Am%29%29 = abs%28%281+-+2m%29%2F%282%2Bm%29%29.


QED.